Finite Group Databasebeta

← #16945371

Subgroups — #16945371

Order 4352.

764 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14532140C211yesmaximal, characteristic
2#14528776C211yesmaximal, characteristic, complemented
3#14526788C214yesmaximal, complemented
4C17×#1459C211yesmaximal, characteristic, complemented
5C23×D68C411yescharacteristic
6C2×#10587616—21no—
7C2×#10587623—21no—
8#11762833—21no—
9C2×#10587504—21no—
10#11762726C2212yes—
11#11762718C412yes—
12#11762711—21no—
13#11762669C411yescharacteristic
14#11762664—21no—
15#11762578C2212yescomplemented
16C2×C4×D4×C17C2211yescharacteristic
17C2×C17×C22⋊C8—21no—
18C17×#324C2212yescomplemented
19C17×#323—21no—
20C24×D17—21no—
21C22×D4×C17D411yescharacteristic
22C23×C4×C17C2×C411yescharacteristic
23C23×C4×C17D411yescharacteristic
24C22×C22⋊D17—22no—
25C22×D68—21no—
26C22×D68—44no—
27C22×D68—41no—
28C22×D68—41no—
29C22×D68C2×C412yes—
30C22×C8×C17—41no—
31C4×D4×C17—41no—
32C4×D4×C17—41no—
33C2×C4⋊C4×C17C2311yescharacteristic
34C2×C22⋊C4×C17C2×C411yescharacteristic
35C2×C22⋊C4×C17D411yescharacteristic
36C2×C42×C17D411yescharacteristic
37C2×C22⋊Dic17—21no—
38#10587623—42no—
39#10587623—42no—
40#10587616—44no—
41C2×#63227—22no—
42C2×#63227—22no—
43C2×#63227—22no—
44C2×C4⋊Dic17—21no—
45C2×#63223—22no—
46C17×C22⋊C8—22no—
47C17×C22⋊C8—41no—
48#10587504—42no—
49#10587502C2×C412yes—
50C4⋊#3622—41no—
51C4⋊#3622—41no—
52C24×C17C22⋊C411yescharacteristic
53C24×C17C22⋊C411yescharacteristic
54C23×D17—22no—
55C23×D17—44no—
56C23×D17—42no—
57C23×D17—42no—
58C23×D17—42no—
59C23×D17—82no—
60C2×D4×C17—41no—
61C2×D4×C17—41no—
62C2×D4×C17D812yes—
63C2×D4×C17SD1612yes—
64C22×C4×C17—21no—
65C22×C4×C17—21no—
66C22×C4×C17—21no—
67C22×C4×C17—21no—
68C22×C4×C17—42no—
69C22×C4×C17—42no—
70C22×C4×C17C22⋊C411yescharacteristic
71C22×C4×C17C22⋊C411yescharacteristic
72C22×C4×C17C22×C411yescharacteristic
73C22×C4×C17C2×D411yescharacteristic
74C22×C4×C17C2×D411yescharacteristic
75C2×C22⋊D17—44no—
76C2×C22⋊D17—44no—
77C2×C22⋊D17—82no—
78C22×Dic17—22no—
79C2×D68—24no—
80C2×D68—42no—
81C2×D68—42no—
82C2×D68—48no—
83C2×D68—44no—
84C2×D68—41no—
85C2×D68—41no—
86C2×D68—44no—
87C2×D68—44no—
88C2×C8×C17—42no—
89C2×C8×C17—82no—
90C4⋊C4×C17—41no—
91C22⋊C4×C17—41no—
92C22⋊C4×C17—41no—
93C42×C17—21no—
94C42×C17—21no—
95C22⋊Dic17—42no—
96#63227—44no—
97#63227—42no—
98#63227—42no—
99#63227—82no—
100C4⋊Dic17—42no—
101#63223—42no—
102#63223—42no—
103C2×#3622—42no—
104#11880—171nomaximal, Sylow
105C23×C17—21no—
106C23×C17—21no—
107C23×C17—21no—
108C23×C17—21no—
109C23×C17—41no—
110C23×C17—41no—
111C23×C17—41no—
112C23×C17—41no—
113C23×C17#10212yes—
114C23×C17#10212yes—
115C23×C17C2×C22⋊C411yescharacteristic
116C22×D17—24no—
117C22×D17—42no—
118C22×D17—42no—
119C22×D17—42no—
120C22×D17—48no—
121C22×D17—42no—
122C22×D17—42no—
123C22×D17—42no—
124C22×D17—88no—
125C22×D17—88no—
126C22×D17—84no—
127C22×D17—84no—
128D4×C17—42no—
129D4×C17—42no—
130C2×C4×C17—21no—
131C2×C4×C17—21no—
132C2×C4×C17—22no—
133C2×C4×C17—21no—
134C2×C4×C17—21no—
135C2×C4×C17—21no—
136C2×C4×C17—42no—
137C2×C4×C17—42no—
138C2×C4×C17—42no—
139C2×C4×C17—41no—
140C2×C4×C17—41no—
141C2×C4×C17—41no—
142C2×C4×C17—81no—
143C2×C4×C17#9911yescharacteristic
144C2×C4×C17#9911yescharacteristic
145C2×C4×C17C2×D811yescharacteristic
146C2×C4×C17C2×SD1611yescharacteristic
147C22⋊D17—88no—
148C2×Dic17—42no—
149C2×Dic17—42no—
150C2×Dic17—42no—
151D68—44no—
152D68—44no—
153D68—44no—
154D68—44no—
155C8×C17—82no—
156#3622—82no—
157#2408—171no—
158#1800—171no—
159#1459D1711yescharacteristic, complemented
160#1345—174no—
161C22×C17—21no—
162C22×C17—21no—
163C22×C17—22no—
164C22×C17—22no—
165C22×C17—42no—
166C22×C17—42no—
167C22×C17—42no—
168C22×C17—41no—
169C22×C17—41no—
170C22×C17—41no—
171C22×C17—41no—
172C22×C17—81no—
173C22×C17C2×#9911yescharacteristic
174C22×C17C2×#10211yescharacteristic
175C22×C17#42111yescharacteristic
176C2×D17—44no—
177C2×D17—44no—
178C2×D17—44no—
179C2×D17—82no—
180C2×D17—82no—
181C2×D17—816no—
182C2×D17—82no—
183C2×D17—82no—
184C4×C17—21no—
185C4×C17—21no—
186C4×C17—21no—
187C4×C17—41no—
188C4×C17—41no—
189C4×C17—41no—
190C4×C17AΣL(1,𝔽16)12yes—
191Dic17—82no—
192C23×D4—171no—
193C2×#121—341no—
194C2×#120—341no—
195C2×C4×D4C2×D1711yescharacteristic
196#423—341no—
197C2×#100—341no—
198C2×C22⋊C8—21no—
199#394—172no—
200#392—172no—
201#390—341no—
202#380—171no—
203#379—341no—
204#343—172no—
205#324C2×D1712yescomplemented
206#323—21no—
207C2×C17—21no—
208C2×C17—21no—
209C2×C17—42no—
210C2×C17—42no—
211C2×C17#151911yescharacteristic
212C2×C17C2×AΣL(1,𝔽16)11yescharacteristic
213C2×C17#213411yescharacteristic
214D17—84no—
215C25—341no—
216C22×D4—172no—
217C22×D4—342no—
218C22×D4—341no—
219C22×D4—681no—
220C22×D4—681no—
221C22×D4—684no—
222C22×D4D6811yescharacteristic
223C23×C4C4×D1711yescharacteristic
224C23×C4C22⋊D1711yescharacteristic
225C2×M16—342no—
226C22×C8—41no—
227#121—684no—
228#120—682no—
229#120—682no—
230C4×D4—41no—
231C4×D4—41no—
232C2×C4⋊C4—341no—
233C2×C4⋊C4C22×D1711yescharacteristic
234C2×C22⋊C4—341no—
235C2×C22⋊C4—342no—
236C2×C22⋊C4—342no—
237C2×C22⋊C4—342no—
238C2×C22⋊C4C4×D1711yescharacteristic
239C2×C22⋊C4C22⋊D1711yescharacteristic
240C2×C42D6811yescharacteristic
241C4⋊C8—681no—
242C4⋊C8—681no—
243#100—682no—
244C22⋊C8—22no—
245C22⋊C8—41no—
246#95—172no—
247C17#1188011yescharacteristic, complemented, Sylow
248C24—342no—
249C24—682no—
250C24—682no—
251C24—684no—
252C24—682no—
253C24—1362no—
254C24#6322711yescharacteristic
255C24#6322711yescharacteristic
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—344no—
259C2×D4—682no—
260C2×D4—682no—
261C2×D4—688no—
262C2×D4—684no—
263C2×D4—684no—
264C2×D4—684no—
265C2×D4—684no—
266C2×D4—684no—
267C2×D4—681no—
268C2×D4—681no—
269C2×D4—1362no—
270C2×D4#6321912yes—
271C2×D4D13612yes—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—342no—
279C22×C4#6322711yescharacteristic
280C22×C4#6322711yescharacteristic
281C22×C4C2×C4×D1711yescharacteristic
282C22×C4C2×D6811yescharacteristic
283C22×C4C2×C22⋊D1711yescharacteristic
284M16—682no—
285M16—682no—
286C2×C8—42no—
287C2×C8—82no—
288C2×C8—682no—
289C4⋊C4—41no—
290C4⋊C4—682no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—684no—
294C22⋊C4—682no—
295C22⋊C4—682no—
296C22⋊C4—682no—
297C22⋊C4—1362no—
298C42—21no—
299C42—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—344no—
309C23—682no—
310C23—682no—
311C23—682no—
312C23—688no—
313C23—682no—
314C23—682no—
315C23—682no—
316C23—1364no—
317C23—1368no—
318C23—1368no—
319C23—1364no—
320C23#1058749212yes—
321C23#1058749212yes—
322C23C2×#6322711yescharacteristic
323D4—42no—
324D4—42no—
325D4—684no—
326D4—684no—
327D4—684no—
328D4—684no—
329D4—1368no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—81no—
343C2×C4—682no—
344C2×C4—682no—
345C2×C4—682no—
346C2×C4#1058747711yescharacteristic
347C2×C4#1058747711yescharacteristic
348C2×C4C2×#6321911yescharacteristic
349C2×C4C2×D13611yescharacteristic
350C8—82no—
351C8—1362no—
352C22—21no—
353C22—21no—
354C22—22no—
355C22—22no—
356C22—42no—
357C22—42no—
358C22—42no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—81no—
364C22—684no—
365C22—684no—
366C22—684no—
367C22—1362no—
368C22—1362no—
369C22—13616no—
370C22—1362no—
371C22—1362no—
372C22C2×#1058747711yescharacteristic
373C22C2×#1058749211yescharacteristic
374C22#1176282511yescharacteristic
375C4—21no—
376C4—21no—
377C4—21no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—1362no—
382C4#1176259512yes—
383C2—21no—
384C2—21no—
385C2—42no—
386C2—42no—
387C2—1364no—
388C2#1452735311yescharacteristic
389C2C2×#1176259511yescharacteristic
390C2#1453047511yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#1459C2
socleC22×C17C2×#99
prime radical#14532140C2
cosocleC2×C4⋊C4×C17C23
hypercenter#1459D17
hypocenterC17#11880
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1459

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #14528776C2semidirect
3. #14526788C2semidirect
4. C17×#1459C2semidirect
15. #11762578C22semidirect
18. C17×#324C22semidirect
159. #1459D17semidirect
205. #324C2×D17semidirect
247. C17#11880semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#145321402
#145287762
#145267882
C17×#14592
#1188017